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1、ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments●Introduction●Thebasicreproductionratio●Threeexamples●Thresholddynamicsinapatchymodel●IntroductionThebasicreproductionratioistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulati
2、on,byatypicalinfectiveindividual.Autonomousepidemicmodels[7,31]SpecificinfectiousdiseasesSexualdiseases[20]Tuberculosisinpossums[13]Denguefever[12]SARS[15,24,33,40]Peopletravelamongcities[1,2]Patchymodels[32,34-36]Periodicfluctuations(contactrates,birthrates,vaccinati
3、onprogram)Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionr
4、isksinmanyothercases.Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation[11].Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortreq
5、uiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulationsuccessfully.Recently,BacaërandGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment[4].Thepurposeofourcur
6、rentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.●ThebasicreproductionratioWec
7、onsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomogeneouscompartments.Letwitheachxi≥0,bethestateofindividualsineachcompartment.Weassumethatthecompartmentscanbedividedintotwotypes:infectedcompartments,labeledbyi=1,...,m,anduninfectedcompartments,label
8、edbyi=m+1,...,n.DefineXstobethesetofalldisease-freestates:Xs:={x≥0:xi=0,∀i=1,...,m}.betheinputrateofnewlyinfectedindividuals
